Question

Astronomers have discovered a planetary system orbiting a star, which is at a distance of 3.9 × 1014 m from the earth. One planet is believed to be located at a distance of 1.5 × 1012 m from the star. Using visible light with a vacuum wavelength of 556 nm, what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

Answer 1

0.00014456 m

Step-by-step explanation:

Wavelength = λ = 556 nm

Planet Distance = Δy = 1.5 × 10¹² m

Star Distance = Δy' = 3.9 × 10¹⁴ m

Diameter of lens = D

Distance between star and planet separated by angle θ

`\Delta y=\Delta y'd \theta \\\Rightarrow d \theta=(\Delta y)/(\Delta y')\\\Rightarrow d \theta=(1.5* 10^(12))/(3.9* 10^(14))=0.00384\ radians`

Angular resolution by Rayleigh criteria

`d\theta=1.22(\lambda)/(D)\\\Rightarrow D=1.22(\lambda)/(d\theta)\\\Rightarrow D=1.22(556* 10^(-9))/(0.00384)\\\Rightarrow D=0.00014456\ m`

∴Diameter aperture is 0.00014456 m